Beam-divergence deconvolution for diffractive imaging

This Rapid Communication presents a method of beam-divergence deconvolution for diffractive imaging. First, the detected diffraction intensity is formulated as a convolution between the diffraction intensity of parallel incident beams and the divergence of an incident beam. It is shown numerically that the convolution causes the reconstructed image to shrink and become blurred. Next, the algorithm of deconvolution used in the iterative Fourier phase retrieval method is applied to the convoluted diffraction intensity deteriorated by quantum noise. Numerical simulations show that the proposed algorithm recovers the deconvoluted diffraction intensity and improves the reconstructed image. Finally, the algorithm is applied to an electron-beam experiment to reconstruct a multiwall carbon nanotube. The results verified that the algorithm reduces the influence of beam divergence.

Figure 1

(Color online) (a) The central area of the simulated diffraction pattern from the model object $\left({\text{a}}^{\prime }\right)$. $\left({\text{a}}^{\prime }\right)$ Structure of the object characterized by four walls. The inset is an enlargement of the central area of the object. The width between walls corresponds to 0.34 nm scaled by 0.095 nm/pixel. [(b) and (c)] The central area of a diffraction pattern convoluted by beam divergence: $\sigma =1.5,6.0$. [$\left({\text{b}}^{\prime }\right)$ and $\left({\text{c}}^{\prime }\right)$] Images of the reconstructed object of diffractive imaging for $\sigma =1.5,6.0$. Diffractive imaging used the hybrid input-output (HIO) algorithm ($\beta =0.5$, 1000 iterations) and the error-reduction (ER) algorithm (1000 iterations) tight support and real-positive constraints and started at a random phase. (d) The central area of the diffraction pattern including both noise and beam divergence: $\sigma =1.5$. $\left({\text{d}}^{\prime }\right)$ The diffraction pattern from direct deconvolution of (d) by the Richardson-Lucy algorithm (5000 iterations).

Figure 2

Two usages of the deconvolution for diffractive imaging. (a) The algorithm is basically the conventional one, which uses a deconvoluted Fourier intensity as a Fourier constraint. (b) In the proposed algorithm, an iterative deconvolution is incorporated into the conventional iterative phase retrieval. The Fourier amplitude constraint uses the root of the deconvoluted intensity. The deconvolution utilizes an object-domain constraint.

Figure 3

(Color online) (a) The graph of $R$-factor dependency on the number of iterations using the proposed algorithm. Three diffraction patterns were inserted at intermediate stages 2000, 5500, and 7000. (b) The diffraction pattern from the application of the proposed algorithm to $I_{\text{noise}}$, $\sigma =1.5$ in Fig. 1d. (c) Comparison of profiles along central vertical lines in Figs. 1a, 1d, 3b. (d) An image reconstructed from (b). Diffractive imaging using HIO with RL (real positive, $\beta =0.5$, 5000 iterations) and ER with RL (5000 iterations) tight support and real-positive constraints and started at a random phase.

Figure 4

(Color online) (a) Distribution of incident beam divergence (3D and contour plot). (b) The raw experimental diffraction pattern with the same scale as in Fig. 1a. $\left({\text{b}}^{\prime }\right)$ A reconstructed image using the raw diffraction pattern. (c) The diffraction pattern after the proposed deconvolution was applied. $\left({\text{c}}^{\prime }\right)$ A reconstructed image using the deconvoluted diffraction pattern. Diffractive imaging using HIO with RL ($\beta =0.5$, 5000 iterations) and ER with RL (5000 iterations); support $2048\times 50\text{pixels}$ real-positive constraints and started at a random phase. Central missing data were calculated using a diffraction pattern.